I’m not a smile but when gifted, I can bring a smile to your face

Take away the rooster from a mixed drink and put what's left behind before a movable barrier, and you get a part of a car or something you should never do to one.

The sharp one goes where I cannot, yet never works alone. I follow every path it makes, until the tear is gone.

They say to take things up one of me, Yet I'm really just a little gap. You might find me in a belt, A tally stick, or a trap. What am I?

First four hold papers tight Last five, a flat piece of wood alright? Together, where copy or cut go Then paste in place like a pro